Consumer

Leti,i∈(1,2, . . . , K) be the set of consumers in the aggregator’s portfolio, where K is the total number of consumers. The total load l_i,t of the i’th consumer in hourt is composed of two parts: inflexible load l_i,t^inf and flexible load l^f_i,t, i.e. l_i,t = l_i,t^inf +l_i,t^f . Since the inflexible load cannot be shifted, I have focused only on the flexible partl_i,t^f . Each consumer is charged the day-ahead price p^s_t for each kilowatt hour of his/her consumed electricity.

3.1.1 Sources of flexibility

The consumer offers the flexibility of consumption of five flexibility sources: washing ma-chines, clothes dryers, dish washers, heat pumps and electric vehicles. All of them have different consumption patterns, i.e. time when the flexibility is offered, the amount of flexi-bility and the time interval for possible shifting of consumption. Based on the characteristics of flexibility sources, the consumers are divided into three clusters:

Figure 3: The algorithm of the simulations reflecting the sequence of forecasting processes and actors’ decisions

• those who offer the flexibility of small home appliances: washing machines, clothes dryers and dish washers (the need to use these appliances is stochastic, the amount of offered flexibility of one appliance is relatively small, appliances are usually used during the day time and the time interval for possible shifting of consumption is medium, 3-6 hours);

• those who offer the flexibility of heat pumps (available only seven months per year, 24 hours per day, the amount of available flexibility is correlated with outside tempera-ture, the time interval for shifting the consumption is relatively shorter, 3 hours);

• and those who offer the flexibility of their electric vehicles (the flexibility is available only at night, the offered amount is relatively large and time period for shifting is the longest, 10 hours).

In the model, the source of flexibility is denoted byj,j ∈(1, . . . , J). The values ofj depend on a particular scenario and the flexibility sources included in the aggregator’s portfolio and J indicates the total number of flexibility sources in the portfolio.

3.1.2 Consumer’s utility

Thei’th consumer’s utility of using an appliance that provides flexibility is denoted by U_i,j,t^a and the cost of using the appliance in hour t is the product of electricity day-ahead price and the amount of energy used in hourt, i.e. p^s_tl^f_i,j,t. Thus, the value of using an appliance in hourt is

V_i,j,t^a =U_i,j,t^a −p^s_tl_i,j,t^f . (1)

Since the focus is on consumption shifting but not on consumption curtailment, the exact utility of using the appliance U_i,j,t^a is not important. For example, if the consumer wants to wash dishes, the satisfaction of clean dishes will be the same and will not depend on the particular time within the allowed time interval for shifting the consumption. This means that if the washing is moved byn,n ∈(0,1, . . . , N), hours, whereN is the maximum number of hours indicating the interval within the consumption can be moved, the utilitiesU_i,j,t^a and U_i,j,t±n^a will be equal. Instead we should analyse the disutility of shifting the consumption

within the indicated period of time. Unlike the utility of using the appliance U_i,j,t^a , the value V_i,j,t^a differs for every hour within the shifting period, because the cost of energy p^s_tl_i,j,t^f depends on consumption hour t. When the consumer optimises his/her consumption schedule, he or she chooses the highest value of using the appliance V_i,j,t^a within the allowed time interval for shifting the consumption. Due to shifting the consumption by n hours to hour t±n, the change in consumption value can be written as

V_i,j,t^a −V_i,j,t±n^a = (U_i,j,t^a −p^s_tl^f_i,j,t)−(U_i,j,t±n^a −p^s_t±nl^f_i,j,t±n). (2) SinceU_i,j,t^a =U_i,j,t±n^a and l_i,j,t^f =l^f_i,j,t±n ⁶, we get

V_i,j,t^a −V_i,j,t±n^a = (p^s_t −p^s_t±n)l_i,j,t^f . (3)

This difference in values can be seen as a disutility of shifting the consumption. However, due to shifting the consumption, the consumer not only incurs higher cost of energy, but also experiences some level of discomfort, for example, uncertainty of the exact time when the dishes are washed. The level of discomfort increases with the increasing number of times when the consumption has been shifted.⁷ To account for the increasing discomfort, I have introduced a compensation factor γ_i,j,t, γ_i,j,t ∈ [1; 2]. Let m_i,j,t be the number of times the aggregator has used the i’th consumer’s flexibility in the whole optimisation period up to and including hour t. γ_i,j,t can be written as

γ_i,j,t = 1 + 1

M_i,jm_i,j,t, (4)

where Mi,j is the total number of times the i’th consumer can offer his or her flexibility in the whole optimisation period. Thus, every time the aggregator uses the flexibility, the consumer’s discomfort and, therefore, compensation factor to the consumer is increasing at a constant rate 1/M_i,j. This means that the compensation factor is 1/M_i,j ×100 percent higher comparing to the previous time of shifting the consumption. The compensation to consumer for shifting his/her consumption for the m_i,j,t’th time can be written as

l^f_i,j,tp^f_i,j,t=l_i,j,t^f (p^s_t−p^s_t±n)γ_i,j,t, (5)

6Here, it is assumed that due to consumption shifting the required amounts of energy are the same in both hours for all flexibility sources including the heat pumps.

7Harbo and Biegel (2012) also argue that contract settlement cost may depend on the flexibility utilisation extent.

where p^f_i,j,t is the flexibility price offered to the i’th consumer for the flexibility source j.

From (5):

p^f_i,j,t = (p^s_t −p^s_t±n)γ_i,j,t. (6)

One one hand, if compensations are too low, consumers have no incentive to offer their flexibility of consumption. On the other hand, if compensations get too high, the aggregator cannot use the offered flexibility because the cost of shifting load exceeds its value. Thus, the higher compensation factor would encourage consumers’ participation, but also would result in lower use of flexibility. In the results section of this study one can see that with the current form of compensation factor only half of the available flexibility is actually used, while the compensation amounts to consumers are very small. Therefore, changing the compensation factor to one or another direction would either diminish already low compensations to consumers or reduce the actual use of flexibility even further.

The presented concept is similar to the proposal by Harbo and Biegel (2012), where they suggest the “N-curtailment contract”. This contract has a limited number of activations, n, and a compensation for curtailment is increasing with the number of activations. Thus, the consumer is compensated progressively with activation. Harbo and Biegel (2012) also propose a fixed reservation payment at x₀ DKK after which follows an activation fee of (x₁, . . . , x_n) for the following n activations.

3.1.3 Consumer’s optimisation problem

The objective function of the i’th consumer offering flexibility of the source j is the min-imisation of the cost of the electricity consumed by providing as much flexibility of the consumption as possible, given by

Ci(l^f_i,j,t) =

T

X

t=1

p^s_tli,t−p^f_i,j,tl^f_i,j,t. (7)

Here, the cost of consumed electricity is equal to the sum of hourly consumed energy at spot prices p^s_tl_i,t less the revenue from the provided flexibility p^f_i,j,tl_i,j,t^f . The consumer has no information about the flexibility price p^f_i,j,t he or she will be offered while making the initial load schedule decision. Therefore, the consumer’s optimisation problem becomes

a simple exercise of finding the lowest electricity spot prices for the time intervals with flexible consumption. After solving this problem, the aggregator is provided with the flexible consumption schedule. Based on this schedule and estimated savings in imbalance payments, the aggregator offers flexibility prices for changing the initial schedule and the consumer minimises the cost by accepting or rejecting the offer for a particular time period.

The consumer’s optimisation problem has several constraints. First, the total consumption consists of inflexible and flexible parts:

l_i,t =l^inf_i,t +l_i,j,t^f . (8)

Second, flexibility can be provided only by certain home appliances, HPs and/or EVs. This means that the amount of flexible consumptionl_i,j,t^f depends on the power of those appliances and the need to use them. In addition, the source of flexibility determines the time interval for possible consumption shifting. For more details on flexibility sources see section 4.2 Data.